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Questions asking for the intuition behind some definition, conjecture, proof etc. In other words, questions designed to improve or to acquire understanding on a conceptual or intuitive level, as opposed to on a technical or formal level. When asking such a question it can be helpful to include a rough description of ones understanding of the subject at hand (on a technical level).

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Intuition behind formal neighborhood and local ring and formal power series

First of all $\mathbb{C[[x,y]]}$ is local because it is only depend on $\mathbb{C}[x,y]_{(x,y)}$ because by definition it is $lim \frac{\mathbb{C}[x,y]}{(X,Y)^n}$ and all things outside $(X,Y)$ are in …
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